Fractal Holiday

For the Museum of Mathematics

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Happen to have any spheres with mirrored surfaces lying around? If so, you can make your own sphere inversion fractals, as today’s gallery of pictures shows. Here’s the basic idea: take three different-colored reflecting balls and arrange them so that each just kisses both of the other two:

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Note the reflections of each ball in the others, and the reflections of reflections, and the reflections of reflections of reflections, and so on, creating a myriad of ever-smaller, self-similar beautiful details—in other words, a fractal.

In three dimensions, you can actually have four spheres each kissing all of the other three; here’s the pattern of reflections you get with four different-colored balls (red, yellow, purple, and silver) arranged in a tetrahedron, looking into the cavity at the center of the arrangement:

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Just the glints of light from an external light source off of such an arrangement of spheres can create a fractal pattern, as most easily seen in this arrangement of three spheres, all silver:

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Here’s a closeup of the most intricate part, using gold balls:

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And here are two photos of a tetrahedral arrangement of gold balls, showing the dramatic difference in what you record with and without using a flash (can you tell which is which?)

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Finally, as a little bonus in thinking about the geometry of spherical reflections, can you tell what shape the lamp reflected in the top of this mirrored sphere is?

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All of the patterns observed here are closely related to circle inversions, of which you can see a beautiful gallery on line. Have fun making your own spherical reflection fractals, and if you create something spectacular, be sure to send a pic to mondays@momath.org — happy reflecting!

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Executive Director, Museum of Mathematics

View more articles by Glen Whitney

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